A-level Mathematics/AQA/MFP3

Series and limits

Two important limits:

${\displaystyle \lim _{x\rightarrow \infty }\left(x^{k}e^{-x}\right)\rightarrow 0}$ for any real number k

${\displaystyle \lim _{x\rightarrow 0}\left(x^{k}\ln {x}\right)\rightarrow 0}$ for all k > 0

The basic series expansions

${\displaystyle (r=0,1,2,\cdots )}$


${\displaystyle e^{x}=1+x+{x^{2} \over 2!}+{x^{3} \over 3!}+{x^{4} \over 4!}+\cdots +{x^{r} \over r!}+\cdots }$

${\displaystyle \sin x=x-{x^{3} \over 3!}+{x^{5} \over 5!}-\cdots +\left(-1\right)^{r}{x^{2r+1} \over (2r+1)!}+\cdots }$

${\displaystyle \cos x=1-{x^{2} \over 2!}+{x^{4} \over 4!}-\cdots +\left(-1\right)^{r+1}{x^{2r} \over (2r)!}+\cdots }$

${\displaystyle (1+x)^{n}=1+nx+{n(n-1) \over 2!}x^{2}+\cdots +\;{\ n \choose r}\;x^{r}+\cdots }$

${\displaystyle (r=1,2,3,\cdots )}$


${\displaystyle \ln(1+x)=x-{x^{2} \over 2}+{x^{3} \over 3}-\cdots +(-1)^{r+1}{x^{r} \over r}+\cdots }$

Improper intergrals

The integral :${\displaystyle \int _{a}^{b}f(x)\,dx\,}$ is said to be improper if

1. the interval of integration is infinite, or;
2. f(x) is not defined at one or both of the end points x=a and x=b, or;
3. f(x) is not defined at one or more interior points of the interval ${\displaystyle a\leq x\leq b}$.

Polar coordinates

A diagram illustrating the relationship between polar and Cartesian coordinates.

${\displaystyle x=r\cos \theta ,\,}$

${\displaystyle y=r\sin \theta ,\,}$

${\displaystyle r^{2}=x^{2}+y^{2},\,}$

${\displaystyle \tan \theta ={y \over x}}$

The area bounded by a polar curve

For the curve ${\displaystyle r=f(\theta ),\,}$ ${\displaystyle \alpha \leq \theta \leq \beta .\,}$

${\displaystyle A=\int _{\alpha }^{\beta }{1 \over 2}r^{2}d\theta \,}$

r must be defined and be non-negative throughout the interval ${\displaystyle \alpha \leq \theta \leq \beta .\,}$

Numerical methods for the solution of first order differential equations

Euler's formula

${\displaystyle y_{r+1}=y_{r}+hf(x_{r},y_{r})\,}$

The mid-point formula

${\displaystyle y_{r+1}=y_{r-1}+2hf(x_{r},y_{r})\,}$

The improved Eular formula

${\displaystyle y_{r+1}=y_{r}+{1 \over 2}(k_{1}+k_{2})\,}$

where

${\displaystyle k_{1}=hf(x_{r},y_{r})\,}$

and

${\displaystyle k_{2}=hf(x_{r}+h,y_{r}+k_{1}).\,}$

Second order differential equations

Euler's identity

${\displaystyle e^{ix}=\cos x+i\sin x,\,x\in {\mathbb {R} }\,}$

When ${\displaystyle x=\pi ,\,}$ substituting into the identity gives

 ${\displaystyle e^{i\pi }=-1\,}$